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Just to add to these answers, a blues walking bass line tends to me much more repetitive and pattern oriented. They're much more likely to play the same major pentatonic pattern over the progression throughout the whole tune.

For Jazz, the lines tend to be more improvisational, different every time around and moving more with the music rather than having a set, rehearsed part.

Of course it doesn't have to be this way, but I would say this is the main difference, as both jazz and blues can have the same progressions.

Like Tim said, these two genres are really close; there are jazz musicians that play the blues and blues musicians that play the jazz.

Usually, the difference is found in the chords. Someone could characterize blues as more 'simple' (without diminishing the genre); a simple blues would have a progression like

C7 C7 C7 C7
F7 F7 C7 C7
G7 G7 C7 C7

And usually, the blues follow this pattern with some small changes.

But jazz songs are different. Here is a jazz progression called rhythm changes:

| Cmaj7 Am7 | Dm7 G7 |
| Em7 A7 | Dm7 G7 |

(this progression was created by George Gershwin for the song I Got Rhythm)

If you play these two progression on piano or guitar, you'll see that they sound different.

It's the same with the walking bass lines. The point of a walking bass line is to express the chords (with the melody and harmony); the same way you hear the difference when you play the chords, is the same way you will hear the difference between the bass lines of a blues and a jazz song.

The crossover between jazz and blues is very blurred. Some tracks are impossible to pigeonhole - apart from saying they're jazz/blues. A walking bass is a walking bass, moving often 4 in the bar from one chord tone to the next appropriate chord tone, usually climbing up or down scale notes with chromatics when needed.

I guess that as blues tends to use predominantly 7ths and 9ths, that would be a clue, as in the bass lines will move more between I - IV - V than anywhere else, whereas jazz tends to incorporate altered chords and usually more of them. So, bass lines in jazz , when heard alone, will meander to more diverse places in more 'unusual' ways.

There have been a couple of questions recently referring to 'what genre'. How much does it really matter ? I'm interested as I play all sorts, and it gets played if it's good , not because of what category someone deigns to put it in.

Walking bass is popular in many genres, as it’s very flexible stylistically. A “walking” line needs to move, rhythmically and harmonically, so you won’t hear steady bass pedal tones, but that doesn’t rule out much. That means that the bassline can easily reflect the usual stylistic tropes of each genre. Blues and jazz share some stylistic elements, like groove, improvisation, and comping, but they often differ in degree or approach.

The bass has an important role in establishing rhythm and harmony, and that’s where the distinctive elements of each genre will show through. As a general rule, jazz tends more towards experimentation and complexity, whereas blues tends more toward tradition and simplicity. Blues tends to stick closer to a simple, 12-bar I–IV–V chord progression, whereas jazz wanders more harmonically, using the ii, iii, and vi chords more. Rhythmically, the blues line is more likely to play straight quarters, while a jazz bass will use more grace notes, ghost notes, and swing.

A big part of the difference comes from the blues genre’s emphasis on expressive melodic improvisation by soloists, rather than rhythmic or textural complexity. This means that everyone except the soloist (usually a lead guitar) plays much simpler music in support of the lead. Thus, a blues bassist is much more likely to play a simple, repetitive arpeggio progression. In contrast, a jazz bassist has more freedom to improvise and play around with rhythm, scales, chromatic passing notes, and so on.